Team:NAU-CHINA/KS Model

Model_Toehold

1. Can we control the amount of CⅠ protein?

The amount of CⅠ protein is controllable. We can change RBS strength to change mRNA translation rate or add degradation tag to change the protein degradation rate.

2. How to explain the effectiveness of the kill switch?

The engineered Bacillus subtilis did not commit suicide in the laboratory and in the intestine of earthworms because CⅠ repressor protein can bind to the specific inhibition binding site of promoter PCⅠ to prevent production of switch RNA. The kill switch cannot be turned on. When they were expelled out of the intestine, CⅠ protein and Trigger RNA were no longer produced and always degraded. As the amount of CⅠ decreased, more and more free inhibition binding sites were exposed, more and more Switch RNA were produced, which can combine with Trigger RNA to produce MazF, making the engineered bacteria commit suicide.

3. Which is the best RBS and degradation tag combination of CⅠ protein?

In the intestine of earthworms, the production of MazF is the key. Due to the possibility of leakage, the production of MazF is not always at zero. We simulated the leakage of MazF in the intestinal environment to find the combination that generated the least MazF leakage. If the combinations were not unique, considering the simplicity of the pathway design, we would choose the one adding the same degradation tag for CⅠ produced both in the laboratory and in the intestine.

1 Abstract

To avoid biological pollution，we modified the gene pathway of Bacillus subtilis by introducing kill switch enabling the bacteria to survive in the intestine and to die immediately outside the intestine. To ensure the feasibility of the kill switch, we built the Kill Switch Model to help design the gene circuit via MATLAB, which combined the degradation rate of CⅠ protein with different strength RBS. After we quantitatively simulated the effect of the kill switch under various combinations,a reasonable one was given. By adding the LVA degradation tag and selecting B0029 RBS, the kill switch can operate effectively.

2 Background

The level of MazF was affected by the RBS and the degradation rate of CⅠ protein, so we chose to adjust the two to find the best combination.

  • Laboratory Culture Phase

Add IPTG to induce the production of enough CⅠ protein, no MazF.

  • In the Intestine of an Earthworm

The accumulated CⅠ protein induced in laboratory degraded, and in the absence of oxygen, Bacillus subtilis expressed CⅠ protein and Trigger RNA, maybe MazF because of leakage.

  • External Environment

When Bacillus subtilis was expelled out of the intestine. CⅠ protein and Trigger RNA were no longer produced and always degraded. As the concentration of CⅠ decreased and the number of free inhibition sites increased, more and more Switch RNA were produced, which can combine with Trigger RNA to produce MazF.

Here are three ideal stages involved in CⅠ protein concentration and MazF.

Stage Laboratory In the Intestine Out of the Intestine
CⅠ + + -
MazF - - +
Note:+ means existence; - means inexistence

3 Model Hypothesis

  • The degradation rate of mRNA and protein is linear function of their amount with a constant coefficient.
  • The mRNA is generated at a constant constitutive transcription rate.
  • The copy number of plasmids is kept as a constant coefficient.
  • Other species such as RNAP polymerases and ribosomes are kept constant as well.
  • Reactions are at equilibrium, steady state, or quasi-steady state[1].
  • Each RBS part has a native strength irrespective of the promoter and protein-coding part it can be used with, and the translation rate with same RBS has a linear relationship with the mRNA length[1].
  • Since the probability of gene mutation is very small, our model did not consider gene mutation.

4 Symbolic Description

Symbol Explanation Value Units
$$CI-1$$ CⅠ protein induced in laboratory
$$CI-2$$ CⅠ protein produced in earthworm intestine
$$ K_{mRNA-CI} $$ The transcriptional rate of DNACI 5.39 $$ min^{-1} $$
$$K_{TR}$$ The transcriptional rate of DNATR 19.12 $$ min^{-1} $$
$$K_{TS}$$ The transcriptional rate of DNATS 2.52 $$ min^{-1} $$
$$K_{CI-2}$$ The translation rate of mRNACI in the intestine of earthworms 4.84 $$ min^{-1} $$
$$K_{tr.ts}$$ The translation rate of generating MazF 2.09 $$ min^{-1} $$
$$d_{mRNA- CI}$$ The degradation rate of mRNACⅠ 0.1386 $$ min^{-1} $$
$$d_{TR}$$ The degradation rate of Trigger RNA 0.0365 $$ min^{-1} $$
$$d_{TS}$$ The degradation rate of Switch RNA 0.18 $$ min^{-1} $$
$$d_{CI-2}$$ The degradation rate of CⅠ produced in earthworm intestine
$$d_{CI-1}$$ The degradation rate of CⅠ induced in laboratory
$$G_{CI}$$ The copy number of CⅠ protein 500
$$G_{TR}$$ The copy number of Trigger RNA 500
$$G_{TS}$$ The copy number of Switch RNA 500
$$V_{mRNA-CI}$$ The formation rate of mRNACI
$$V_{TR}$$ The formation rate of Trigger RNA
$$V_{TS}$$ The formation rate of Switch RNA
$$ \varepsilon _{1} $$ The coefficient of CⅠ protein and inhibitory site binding
$$O_{CI-1}$$ The first inhibition site of PCⅠ
$$O_{CI-2}$$ The second inhibition site of PCⅠ
$$K_{1}$$ The binding constant of Trigger RNA and Switch RNA $$1\times10^{5}$$
$$K_{CIo1}$$ The equilibrium associationconstant between the repressor and OCⅠ-1inhibition sites $$1\times10^{11\quad[5]}$$ $$M^{-1}$$
$$K_{CIo2}$$ The equilibrium associationconstant between the repressor and OCⅠ-1inhibition sites $$1\times10^{9\quad[5]}$$ $$M^{-1}$$
$$K_{CId}$$ The equilibrium association constant between the repressor and other sites $$8.3\times10^{6\quad[5]}$$ $$M^{-1}$$
$$\lambda _{12}$$ The ratio of site 1 by DNA looping of repressor and site 2 complex 100
Note: 1. Calculation method of transcription rate: The general transcription rate is 40~80 nt/s[2]. We took 80nt/s as the standard to calculate the transcription rate according to the length of DNA sequence.
Note: 2. Calculation method of translation rate: The general translation rate range is 10~20 aa/s[2]. Except for CⅠ proteins, 20 aa/s was used as the standard to calculate the translation rate according to the mRNA sequence length. The CⅠ protein translation rate at the highest RBS strength was set as 20 aa/s, and the translation rates at other RBS strengths were calculated in turn according to the intensity ratio.

5 Design

By regulating the amount of CⅠ protein, we can control the kill switch. In the laboratory, we selected the concentration of IPTG which could produce the most CⅠ protein. After entering the intestine, RBS strength and CⅠ protein degradation rate were adjusted by using our model to achieve the minimum amount of MazF. In addition, the combination should meet the need that the engineered bacteria can commit suicide as soon as possible when they were expelled out of the intestine.

5.1 RBS and Degradation Tag

RBS Selection

RBS sequence, also called SD ( Shine-Dalgarno ) sequence, is a key controlling the initiation of translation and the expression of proteins, so it determines the level of translation, increasing yield of target product [3]. We can change the amount of CⅠ protein produced by selecting suitable RBS strength.

To select a more suitable RBS, we selected RBS with different strengths for the test, as shown in Table 5.1.1.

Table 5.1.1 Four types of RBS
Name Translation rate(min-1)
B0034 4.84
B0064 3.02
B0029 0.52
B0033 0.04
Note: Calculation method of translation rate: The general translation rate range is 10~20 aa/s[2]. The CⅠ protein translation rate at the highest RBS strength was set as 20 aa/s, and the translation rates at other RBS strengths were calculated in turn according to the intensity ratio.
CⅠ Degradation Tags

The exact rate of protein degradation depends on a number of factors: the concentration of CLPXP and ClpAP protease and SspB medium; the stability of the protein; the km with the protease binds; the temperature, etc. But we can change the rate of degradation by adding tags. The tags can be recognized by the CLPX foldase and forms a complex with the ClpP protease. The last three residues of the tag determine the strength of the interaction with Clpx, thus determining the ultimate protein degradation rate.

We chose three tags: LVA, AAV and ASV. Under the LVA tag, the degradation rate of CⅠ reached 0.1733 min-1 , and the degradation rate reached 0.0173 min-1 and 0.0087 min-1 due to AAV and ASV tag[4].

Bacillus subtilis expressed CⅠ protein both in the laboratory and in the intestine of earthworms. We can control the degradation rate of both stages at the same time, and choose two degradation tags for the two stages. Respectively, we got some different degradation rate combinations, as shown in Table 5.1.2.

Table 5.1.2 C1 degradation rate combinations
CⅠ-2 CⅠ-1 LVA LVA AAV AAV ASV ASV 0.1733 0.0173 0.0087 0.1733/0.0173 0.0173/0.1733 0.012/0.0062 0.0087/0.1733 0.0087/0.0173 0.1733/0.0087

5.2 CⅠ Rules

Since CⅠ proteins were produced to form dimer and inhibitory site binding, in this section we had formulated rules for such binding, with CⅠ2 representing the dimer CⅠ repressor and OCⅠ-1 and OCⅠ-1 representing the two CⅠ suppressor sites[5]

Dimer CⅠ repressor

$$ CI2_{f}+O_{CⅠ-1f}\rightleftharpoons CI2:O_{CI-1} $$ $$ CI2_{f}+O_{CⅠ-2f}\rightleftharpoons CI2:O_{CI-2} $$ $$ CI2:O_{CI-1}+O_{CI-2f}\rightleftharpoons O_{CI-1}:CI2:O_{CI-2}\rightleftharpoons CI2:O_{CI-2}+O_{CI-1f} $$ $$ CI2_{f}+D\rightleftharpoons CI2:D $$

Operating area 1

$$ CI2_{f}+O_{CI-1f}\rightleftharpoons CI2:O_{CI-1} $$ $$ CI2:O_{CI-1}+O_{CI-2f}\rightleftharpoons O_{CI-1}:CI2:O_{CI-2}\rightleftharpoons CI2:O_{CI-2}+O_{CI-1f} $$

Operating area 2

$$ CI2_{f}+O_{CI-2f}\rightleftharpoons CI2:O_{CI-2} $$ $$ CI2:O_{CI-1}+O_{CI-2f}\rightleftharpoons O_{CI-1}:CI2:O_{CI-2}\rightleftharpoons CI2:O_{CI-2}+O_{CI-1f} $$

Relationship:

$$ \left [ CI2\right ]=\left [ CI2_{f}\right ]+\left [ CI2:O_{CI-1}\right ]+\left [ CI2:O_{CI-2}\right ]+\left [ O_{CI-1}:CI2:O_{CI-2}\right ]+\left [ CI2:D\right ] $$ $$ \left [ O_{CI-1}\right ]=\left [ O_{CI-1_{f}}\right ]+\left [ CI2:O_{CI-1}\right ]+\left [ O_{CI-1}:CI2:O_{CI-2}\right ] $$ $$ \left [ O_{CI-2}\right ]=\left [ O_{CI-2_{f}}\right ]+\left [ CI2:O_{CI-2}\right ]+\left [ O_{CI-1}:CI2:O_{CI-2}\right ] $$

Where:

$$ \left [ CI2:O_{CI-1}\right ]=K_{CIo1}\cdot \left [ CI2_{f}\right ]\cdot \left [ O_{CI-1_{f}}\right ] $$ $$ \left [ CI2:O_{CI-2}\right ]=K_{CIo2}\cdot \left [ CI2_{f}\right ]\cdot \left [ O_{CI-2_{f}}\right ] $$ $$ \left [ O_{CI-1}:CI2:O_{CI-2}\right ]=K_{CIo2}\cdot \lambda _{12}\left [ CI2:O_{CI-1}\right ]\cdot \left [ O_{CI-2_{f}}\right ]=K_{CIo1}\cdot\left [ CI2:O_{CI-2}\right ]\cdot \left [ O_{CI-1_{f}}\right ] $$ $$ \left [ CI2；D\right ]=K_{CId}\cdot \left [ CI2_{f}\right ]\cdot \left [ D_{f}\right ] $$ $$ \left [ D\right ]=\left [ CI2:D\right ]+\left [ D_{f}\right ] $$

Note：CⅠ2 represents the dimer CⅠ repressor; OCⅠ-1 and OCⅠ-2 represent the two CⅠ inhibition site; Af representsA is free; A:B means A binds to B inhibition site; B:A:C means B and C inhibitionsites are both occupied; KCⅠo.. is the equilibriumassociation constant between the repressor and sites; λ12 means the ratio of site 1 by DNA looping of repressorand site 2 complex; D means other binding sites;[·] means concentration.

6 Modeling

6.1 In the Intestine

Constitutive expression, not affected by time, place, or environment, has no spatiotemporal specificity. The gene that encodes the protein is not dependent on any transcription factor, so it will continuously transcribe mRNA molecules, and then the translation of the protein will be out of control[3]. The expression of CⅠ protein in the intestine of earthworm can be easily described in Figure 6.1. The process of generating Switch RNA and Trigger RNA was similar to the process of producing mRNACⅠ in the figure.

Fig.6.1. Production and Inhibition Effect of CⅠ Protein

The CⅠ protein, Switch RNA and Trigger RNA expression can be simply translated into the following biochemical reactions,

$$DNA_{CI}\overset{K_{mRNA-CI}}{\rightarrow} mRNA_{CI}$$ $$mRNA_{CI}\overset{K_{CI}}{\rightarrow}CI$$ $$mRNA_{CI}\overset{K_{mRNA-CI}}{\rightarrow}\phi$$ $$CI\overset{d_{CI}}{\rightarrow}\phi$$ $$DNA_{TR}\overset{K_{TR}}{\rightarrow} Trigger\quad RNA$$ $$Trigger\quad RNA\overset{d_{TR}}{\rightarrow}\phi$$ $$DNA_{TS}\overset{K_{TS}}{\rightarrow} Switch\quad RNA$$ $$Switch\quad RNA\overset{d_{TS}}{\rightarrow}\phi$$

6.1.1 CⅠ Protein Production

Here, we used ordinary differential equation to calculate the concentration of each substance. The concentration of mRNACⅠ can be expressed as follows:

$$\frac{\mathrm{d}\left[mRNA_{CI}\right]}{\mathrm{d}t}={V_{mRNA-CI}}-\left [ mRNA_{CI}\right ]d_{mRNA-CI}$$

Where, dmRNA-CⅠ(0.1386min-1) is the degradation rate of mRNACⅠ and VmRNA-CⅠ is the generation rate of mRNACⅠ, which can be expressed as follows:

$$ V_{mRNA-CI}= K_{mRNA-CI}\left [ G_{CI}\right ] $$

Where, KmRNA-CⅠ is the transcription rate, and KmRNA-CⅠ = 5.39 min -1, [GCⅠ] is the copy number of plasmids.

Here's what follows:

$$\frac{\mathrm{d}\left [ CI\right ]}{\mathrm{d}t}=V_{CI}-\left [ CI-2\right ]d_{CI-2}-\left [ CI-1\right ]d_{CI-1}$$

Here, dCⅠ-1 is the laboratory-induced degradation rate of CⅠ protein, dCⅠ-2 is the degradation rate of CⅠ protein produced in earthworm intestines, both of which can be adjusted, and VCⅠ is the production rate:

$$V_{CI}=K_{CI-2}\left [ mRNA_{CI-2}\right ]$$

KCⅠ-2=4.84 min -1 is the translation rate of CⅠ generated in the intestine, which could be changed by adjusting RBS. The quantities of CⅠ-1 and CⅠ-2 in the intestine are given by model_CⅠ_inner_function. The total amount of CⅠ in intestine is shown in Figure 6.1.1.

Fig.6.1.1. The Total Amount of CⅠ in Intestine

6.1.2 Trigger RNA Production

Similarly, the concentration of Trigger RNA in the intestine can be calculated:

$$\frac{d\left [ TR\right ]}{dt}=V_{TR}-\left [ TR\right ]d_{TR}$$

Where dTR is constant, dTR = 0.0365 min-1, and VTR is generation rate of mRNATR, given by the following formula:

$$V_{TR}=K_{TR}\left [ G_{TR}\right ]$$

KTR=19.12min-1, [GCⅠ] =500 is the copy number of plasmids.

The amount of Trigger RNA in the intestine is shown in Figure 6.1.2.

Fig.6.1.2. The Amount of Trigger RNA in the Intestine

6.1.3 Switch RNA Production

In order to solve the concentration problem of Switch RNA, we also assumed:

$$\frac{d\left [ TS\right ]}{dt}=V_{TS}-\left [ TS\right ]d_{TS}$$

Where dTSis constant, dTS= 0.18 min -1and VTS is generation rate of Switch RNA，given by the following formula:

$$V_{TS}=K_{TS}\varepsilon _{1}\left [ G_{TS}\right ]$$

KTS= 2.52 min -1, [GCⅠ] =500, ε1 indicates the binding of CⅠ and inhibition sites, which can be expressed as follows and calculated by model_calculate_1:

$$\varepsilon _{1}=\left ( \frac{\left [ O_{CI-1f}\right ]}{\left [ O_{CI-1}\right ]}\right )\left ( \frac{\left [ O_{CI-2f}\right ]}{\left [ O_{CI-2}\right ]}\right )$$

The amount of Switch RNA in the intestine is shown in Figure 6.1.3, which can be calculated by model_Toehold_function.

Fig.6.1.3. The Amount of Switch RNA in Intestine

6.1.4 MazF Production

The expression of MazF in the intestine of earthworm can be easily described in Figure 6.1.4.1.

Fig.6.1.4.1. The Production of MazF

The specific process of generating MazF is represented by the following formula:

$$\left [trRNA \right ]+\left [ tsRNA\right ] \overset{K_{1}}{\rightleftharpoons} \left [ tr.tsRNA\right ]$$

$$\left [ tr.tsRNA\right ]\overset{K_{tr.ts}}{\rightarrow} \left [ mazF\right ]$$

Among them, Ktr.ts = 2.09min -1 is the translation rate of generating MazF. K1 = 1*10 5，is the reaction equilibrium constant described as follows:

$$K_{1}=\frac{\left [ tr.tsRNA\right ]}{\left [ trRNA\right ]\left [ tsRNA\right ]}$$

The total amount of Trigger RNA and Switch RNA are shown as follows:

$$\left [ T_{tr}\right ]=\left [ trRNA\right ]+\left [ tr.tsRNA\right ]$$

$$\left [ T_{ts}\right ]=\left [ tsRNA\right ]+\left [ tr.tsRNA\right]$$

We built two functions, model_calculate_2 and model_MazF_function, to calculate the quantities of tr.tsRNA and MazF.

The amount of MazF in the intestine is shown in Figure 6.1.4.2.

6.1.4.2
Fig.6.1.4.2. The Amount of MazF in the Intestine

6.2 Out of the Intestine

After the engineered bacteria were expelled out of the intestine, Trigger RNA and CⅠ protein were no longer produced and always degraded. Switch RNA was produced in the same way as in the intestine. And with the gradual degradation of CⅠ, Trigger RNA and Switch RNA would combine to produce MazF. The expression of CⅠ protein, Trigger RNA, Switch RNA and MazF were obtained by applying differential equations above, as shown below.

6.2.1
Fig. 6.2.1．The Total Amount of CⅠ out of the Intestine
6.2.2
Fig.6.2.2. The Amount of Trigger RNA out of the Intestine
6.2.3
Fig.6.2.3. The Amount of Switch RNA out of the Intestine
6.2.4
Fig.6.2.4．The Amount of MazF out of the Intestine

7 Results & Analysis

MazF directly determined whether the engineered bacteria will commit suicide or not. In order to ensure the kill switch to work, we analyzed the concentration of MazF under different combinations in and out of the intestine Figure 7.1.

7.1
Fig.7.1. MazF Under Different Combinations in and out of the Intestine

We found that under the threshold set by us, only three combinations can meet the requirement of MazF generated in earthworm intestines less than the threshold of suicide, and far less than MazF generated out of the intestine. In addition, MazF generated out of the intestine is higher than the threshold, which can make the engineered bacteria commit suicide.

Considering the simplicity of the pathway design, we hope to add the same degradation tag for CⅠ-1 and CⅠ-2. Therefore, our choice was: B0029 RBS and LVA degradation tag.

Under this combination, the quantities of various substances are shown in Figure 7.2.

7.2
Fig.7.2. Various Substances Concentration

8 Sensitivity Analysis

We had known that the promoter strength can affect the transcription rate. In fact, data related to promoter strength is difficult to obtain. Therefore, sensitivity analysis was performed on promoter strength, making the transcription rate range from 2 min-1 to 6 min-1, as shown in Figure 8.1.

fig.8.1
Fig.8.1. Sensitivity Analysis of Transcription Rate.

The engineered bacteria were induced to produce CⅠ proteins by IPTG in the laboratory. We had selected the concentration of IPTG that could produce the most CⅠ proteins to inhibit the production of MazF. And in order to explore the influence of CⅠ-1 concentration on the subsequent generation of CⅠ-2 concentration, we conducted a sensitivity analysis on the CⅠ protein concentration of engineered bacteria when they entered the earthworm intestine, as shown in Figure 8.2.

fig.8.2.
Fig.8.2. Sensitivity Analysis of Initial Concentration.

The results showed that:

  • The CⅠ-2 protein production was sensitive with the promoter strength. But since the promoter strength is easy to change, we could only choose the best RBS strength.
  • The CⅠ-2 protein production wasn't sensitive with quantity of CⅠ induced in lab. Thus, our choice on the quantity of CⅠ induced in lab is reasonable.

References